Vertex operator algebras generated by two Ising vectors

In this paper we mainly study the vertex operator algebra $\mathbb{C} \mathrm{VA}(e, f)$ generated by two Ising vectors $e$ and $f$ with $\langle e, f\rangle=\frac{5}{2^{10}}$. We prove $\mathbb{C} \mathrm{VA}(e, f)$ is isomorphic to the $6A$-algebra $\mathcal{U}_{6A}$ constructed by Lam-Yamada-Yamauchi. We also discuss the cases $\langle e, f\rangle=\frac{1}{4}, \frac{1}{2^{5}}, 0$ and the case $\langle e, f\rangle=\frac{1}{2^{8}}$ with $(\tau_{e}\tau_{f})^4=1$.